#8071356

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6)(7,12)(8,13)(9,14)(10,11)(15,19)(16,17)(18,20)(21,22)(23,26)(24,28)(25,29)(27,30)(31,34)(32,33)(35,36);(1,2,3,4)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,35)(33,36);(6,11)(9,19)(12,17)(13,20)(23,30)(26,27)(31,36)(34,35);(5,7)(6,12)(8,14)(9,13)(10,16)(11,17)(15,18)(19,20)(21,22)(23,26)(24,32)(25,29)(27,30)(28,33)(31,35)(34,36);(5,8)(6,13)(7,15)(9,17)(10,18)(11,20)(12,19)(14,16)(21,23)(22,27)(24,31)(25,30)(26,29)(28,35)(32,34)(33,36);(5,9,10,19)(6,14,11,15)(7,13,16,20)(8,17,18,12)(21,24)(22,28)(23,31)(25,33)(26,34)(27,35)(29,32)(30,36)

Polycyclic presentation:

2^9|b^2=g,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7978552);